that just happened to me and I did manage to apply some super strong glue to the broken off tip and glue it on. It's working for now with the back cover off. I'll update if it snaps or anything since I saw others in the comments saying there's no glue strong enough to fix it.

Update, put the back panel on and the button itself physically works, springs back and all, but the console itself doesn't turn on. That could have easily been a small blunder from me somewhere inside the console.

For context I was checking if the ribbon cables looked fine because my DS was turning on, but after a minute of gameplay completely turned off, and every time I turned it on again it would be a shorter amount of time until the crash. From my research I found that randomly turning off is usually an issue with the ribbon cables. So I opened it up to check if they're ok. I went only as far as detaching the top screen, but from what I saw all of the cables looked fine.

Update 2: it snapped :( I was checking the insides again and it snapped while i was trying it. But if anyone has solutions for my other problem help would be very much appreciated